I'd have to say:
Angel Beats
Elfien Lied
Clannad AS
Myself:Yourself
Air

I just finished watching Anohana and I shed an ungodly amount of manly tears. That last episode.... I'm still trying to fight off the tears but no luck there
Same thing with the other animes I listed. They each are able to perfectly pull on one's heart strings, especially ones that have characters that are so relatable and you can put yourself in their shoes. There are still plenty more I want to add on to my list.

Angel Beats: You can't the end without crying, and then when you start to think about it, if a season 2 was created for this, all the sad parts would be fixed, or atleast 1 last bonus episode, where everyone meet eachother on the other side.

Clannad: Do i even need to explain?

Darker Than Black Trilogy I honestly love this anime, and i hate the way it ends.

Ugh.. Saikano made me feel sick. Every episode was just so depressing, and then the end.. just, ugh. For awhile I told people the anime didn't exist and that I didn't want to talk about it. :'[

I'd say Kanon tugged on my heart strings the most though. Angel Beats was amazing, but.. it had humor and death was taken a little lighter in it. Kanon.. when things happened.. that was it. No extra source of power.. no "strength from friends". Just.. hey, give me your heart. And they bit it, chewed it, stretched it, and when they gave it back.. they wrapped it up like a gift with a chocolate chip cookie -brand new.. and the tears. I could fill up a cup with tears of sadness and tears of laughter. Just.. ugh! Sad. :[

Saikano.. what were they thinking when they made that anime? I have a hard time watching anything with cheating.. School Days and such.. and.. not to spoil anything, but.. I was so angry throughout that anime. The only part I was okay with, was

Spoiler Alert! Click to show or hide

when the main character spent time with Akemi in the hospital after the earthquake. He told her lies so that her death would be easier.. she was happy with her last breath. That was really sweet, but that was the one part in the entire anime that was okay to me.